We are to maximise
Z = 3x + 4 y
subject to the constraints
x + y ≤ 4
x ≥ 0, y ≥ 0
Consider a set of rectangular cartesian axes OXY in the plane.
It is clear that any point which satisfies x ≥ 0, y ≥ 0 lies in the first quadrant.
Let us draw the graph of x + y = 4
For x = 0, y = 4
For y = 0, x = 4
∴ line meets OX in A(4, 0) and OY in L(0, 4)

Since feasible region is the region which satisfies all the constraints.
∴  OAL is the feasible region. The comer points are O(0, 0), A(4, 0), L(0, 4).
At O(0, 0), Z = 0 + 0 = 0
At A(4, 0), Z = 12 + 0 = 12
At L(0, 4), Z = 0 + 16 = 16
∴ maximum value = 16 at (0, 4).

We are to maximise
Z = 4x + y
subject to the constraints
x + y ≤ 50
3x + y ≤ 90
x ≥ 0, y ≥ 0
Consider a set of rectangular cartesian axes OXY in the plane.
It is clear that any point which satisfies x ≥ 0, y ≥ 0 lies in the first quadrant.
Now we draw the graph of the line x + y = 50
For x = 0, y = 50
For y = 0, x = 50
∴ line meets OX in A(50, 0) and OY in L(0, 50)
Let us draw the graph of line 3 x + y = 90
For x = 0, y = 90
For y = 0, 3x = 90 or x = 30
∴ line meets OX in B(30, 0) and OY in M(0, 90).
Since feasible region is the region which satisfies all the constraints.
∴  OBCL is the feasible region, which is bounded.

Tips: -

Note: Coordinates of C can be found by two methods:
Method I: Draw the graph of inequalities on the graph paper. So coordinates of C can be determined.
Method II: Solve the two equation x + y = 50, 3x + y = 90 by any method to find coordinates of C.

We are to maximize
f = x + 2y
subject to the constraints
2x + 3 y ≤ 6
x + 4 y ≤ 4
x, y ≥ 0
Consider a set of rectangular cartesian axes OXY in the plane.
It is clear that any point which satisfies x ≥ 0, y ≥ 0 lies in the first quadrant.
Now we draw the graph of the line 2 x + 3 y = 6.
For x = 0, 3 y = 6, or y = 2
For y = 0, 2 x = 6, or x = 3
∴  line meets OX in A (3, 0) and OY in L (0, 2)
Let us draw the graph of line x + 4 y = 4
For x = 0, 4 y = 4, or y = 1
For y = 0, x = 4
∴  line meets OX in B (4, 0) and OY in M (0, 1)

Since feasible region is the region which satisfies all the constraints
∴    OACM is the feasible region. The comer points are
     
At 

We have to maximize
z = 9x + 3 y
subject to the constraints
2x + 3 y ≤ 13
2x + y ≤ 5
x, y ≥ 0
Consider a set of rectangular cartesian axes OXY in the plane.
It is clear that any point which satisfies x ≥ 0,y ≥ 0 lies in the first quadrant.
Let us draw the graph of 2x + 3y = 13
For x = 0,  3y = 13  
For y = 0,  2x = 13    

Again we draw the graph of 2x + y = 5
For x = 0,  y = 5
For y = 0,  2x = 5  


Since feasible region satisfies all the constraints.
    OCEB in the feasibe region. The corner points are O(0, 0),   
       At O(0, 0),  z = 9(0) + 3(0) = 0+ 0 = 0
At   
At